Positive configurations of flags in a building and limits of positive representations

  • Giuseppe MartoneEmail author


Parreau compactified the Hitchin component of a closed surface S of negative Euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of \(\pi _1(S)\) on an \({\mathbb {R}}\)-Euclidean building. In this paper, we use the positivity properties of Hitchin representations introduced by Fock and Goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building.



It is a pleasure to thank my thesis advisor, Francis Bonahon, for encouraging me to think about this problem, for the numerous insightful conversations, and for his support. I thank Daniele Alessandrini and Beatrice Pozzetti for useful discussions and feedback. I am very grateful to the referee for providing several useful comments on an earlier version of this manuscript.


  1. 1.
    Alessandrini, D.: Tropicalization of group representations. Algebr. Geom. Topol. 8(1), 279–307 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bestvina, M.: Degenerations of the hyperbolic space. Duke Math. J. 56(1), 143–161 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonahon, F.: The geometry of Teichmüller space via geodesic currents. Invent. Math. 92(1), 139–162 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonahon, F.: Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form. Ann. Fac. Sci. Toulouse Math. 5(2), 233–297 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonahon, F., Dreyer, G.: Parameterizing Hitchin components. Duke Math. J. 163(15), 2935–2975 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonahon, F., Dreyer, G.: Hitchin characters and geodesic laminations. Acta Math. 218(2), 201–295 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972)CrossRefzbMATHGoogle Scholar
  8. 8.
    Burger, M., Iozzi, A., Parreau, A., Pozzetti, M.B.: A structure theorem for geodesic currents and length spectrum compactifications. Preprint arxiv:1710.07060 (2017)
  9. 9.
    Burger, M., Pozzetti, M.B.: Maximal representations, non-Archimedean Siegel spaces, and buildings. Geom. Topol. 21(6), 35393599 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cooper D., Delp K., Long D., Thistlethwaite M.: (in preparation) Google Scholar
  11. 11.
    Fathi, A., Laudenbach, F., Poénaru, V.: Thurston’s work on surfaces, volume 48 of Mathematical Notes. Princeton University Press, Princeton, NJ. Translated from the 1979 French original by Djun M. Kim and Dan Margalit (2012)Google Scholar
  12. 12.
    Fock, V., Goncharov, A.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. 103, 1–211 (2006)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gaiotto, D., Moore, G.W., Neitzke, A.: Spectral networks and snakes. Ann. Henri Poincaré 15(1), 61–141 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gantmacher, F.P., Krein, M.G.: Oscillation matrices and kernels and small vibrations of mechanical systems. In: AMS Chelsea Publishing, Providence, RI, revised edition. Translation based on the 1941 Russian original, Edited and with a preface by Alex Eremenko (2002)Google Scholar
  15. 15.
    Gromov, M.: Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2 (Sussex, 1991), volume 182 of London Math. Soc. Lecture Note Ser., pages 1–295. Cambridge Univ. Press, Cambridge (1993)Google Scholar
  16. 16.
    Hitchin, N.J.: Lie groups and Teichmüller space. Topology 31(3), 449–473 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kapovich, M., Leeb, B.: On asymptotic cones and quasi-isometry classes of fundamental groups of \(3\)-manifolds. Geom. Funct. Anal. 5(3), 582–603 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kleiner, B., Leeb, B.: : Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes Études Sci. Publ. Math. 86, 115–197 (1998). (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Labourie, F.: Anosov flows, surface groups and curves in projective space. Invent. Math. 165(1), 51–114 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Le, I.: Higher laminations and affine buildings. Geom. Topol. 20(3), 1673–1735 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Le, I.: Intersection pairings for higher laminations. Preprint arxiv:1708.00780 (2017)
  22. 22.
    Lusztig, G.: Total positivity in reductive groups. In: Lie theory and geometry, volume 123 of Progr. Math., pages 531–568. Birkhäuser Boston, Boston, MA (1994)Google Scholar
  23. 23.
    Morgan, J.W., Shalen, P.B.: Valuations, trees, and degenerations of hyperbolic structures. I. Ann. Math. 120(3), 401–476 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer, Berlin, third edition (1994)Google Scholar
  25. 25.
    Parreau, A.: Immeubles affines: construction par les normes et étude des isométries. In Crystallographic groups and their generalizations (Kortrijk, 1999), volume 262 of Contemp. Math., pages 263–302. Am. Math. Soc., Providence, RI (2000)Google Scholar
  26. 26.
    Parreau, A.: Compactification d’espaces de représentations de groupes de type fini. Math. Z. 272(1–2), 51–86 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Parreau, A.: Invariant subspaces for some surface groups acting on A2-euclidean buildings. Preprint arxiv:1504.03775 (2015)
  28. 28.
    Parreau, A.: On triples of ideal chambers in A2-buildings. Preprint arxiv:1504.00285 (2015)
  29. 29.
    Paulin, F.: Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94(1), 53–80 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Schoenberg, I.: Über variationsvermindernde lineare Transformationen. Math. Z. 32(1), 321–328 (1930)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Thurston, W.P.: Minimal stretch maps between hyperbolic surfaces. Preprint arxiv:math/9801039 (1986)
  32. 32.
    Thurston, W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. (N. S.) 19(2), 417–431 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    van den Dries, L., Wilkie, A.J.: Gromov’s theorem on groups of polynomial growth and elementary logic. J. Algebra 89(2), 349–374 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA

Personalised recommendations